(*
  DaoMath Conservation Theorems
  守恒定理的形式化证明
*)

Require Import Reals.
Require Import InnateBalance.
Require Import Quantum.
Open Scope R_scope.

(** * 能量守恒 *)

Parameter energy : QuantumState -> R.

Axiom energy_conservation_quantum :
  forall (psi : QuantumState) (U : Operator) (t : R),
  is_unitary U ->
  energy (apply_operator U psi) = energy psi.

Theorem energy_conservation_evolution :
  forall (psi : QuantumState) (H : Operator) (t : R),
  is_hermitian H ->
  energy psi = energy (apply_operator H psi).
Proof.
  intros psi H t H_herm.
  (* 使用 unitary 演化 *)
Admitted.

(** * 概率守恒 *)

Theorem probability_conservation :
  forall (psi : QuantumState) (A : Operator),
  is_normalized psi ->
  is_hermitian A ->
  measurement_probability psi A <= 1.
Proof.
  intros psi A H_norm H_herm.
  apply born_rule; assumption.
Qed.

(** * 信息守恒（幺正演化） *)

(** von Neumann 熵 *)
Parameter von_neumann_entropy : QuantumState -> R.

Axiom entropy_unitary_invariance :
  forall (psi : QuantumState) (U : Operator),
  is_unitary U ->
  von_neumann_entropy (apply_operator U psi) = 
  von_neumann_entropy psi.

Theorem information_conservation :
  forall (psi : QuantumState) (U : Operator),
  is_normalized psi ->
  is_unitary U ->
  von_neumann_entropy psi >= 0.
Proof.
  intros psi U H_norm H_unitary.
  (* von Neumann 熵非负性 *)
Admitted.

(** * 对偶守恒 *)

Theorem duality_conservation :
  forall (x : R),
  x + dual x = 0.
Proof.
  intros x.
  unfold dual. simpl. lra.
Qed.

(** 推广到量子态 *)
Parameter quantum_dual : QuantumState -> QuantumState.

Axiom quantum_duality : forall psi : QuantumState,
  inner_product psi (quantum_dual psi) = mkComplex 0 0.

